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By William Henry Besant

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Low Dimensional Topology

During this quantity, that's devoted to H. Seifert, are papers in keeping with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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10. If SE be the perpendicular from the focus on the normal at P , shew that SE 2 = AN . SP. 11. The locus of the vertices of all parabolas, which have a common focus and a common tangent, is a circle. 12. Having given the focus, the length of the latus rectum, and a tangent, construct the parabola. 13. If P SP be a focal chord, and P N , P N the ordinates, shew that AN . AN = AS 2 . Shew also that the latus rectum is a mean proportional between the double ordinates. 14. The locus of the middle points of the focal chords of a parabola is another parabola.

Two equal parabolas have a common focus; and, from any point in the common tangent, another tangent is drawn to each; prove that these tangents are equidistant from the common focus. 112. Two parabolas have a common axis and vertex, and their concavities turned in opposite directions; the latus rectum of one is eight times that of the other; prove that the portion of a tangent to the former, intercepted between the common tangent and axis, is bisected by the latter. CHAPTER III. The Ellipse. Def.

2. Draw a tangent to a parabola, making a given angle with the axis. 3. If the tangent at P meet the tangent at the vertex in Y , AY 2 = AS . AN. 4. If the normal at P meet the axis in G, the focus is equidistant from the tangent at P and the straight line through G parallel to the tangent. 5. Given the focus, the position of the axis, and a tangent, construct the parabola. 6. Find the locus of the centre of a circle which touches a given straight line and a given circle. 7. Construct a parabola which has a given focus, and two given tangents.

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